arXiv research

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…

We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Ampère equations. This extends to a new family of mixed type examples various classi…

Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle

Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of …

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existen…

We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characterist…

Study harmonicity of normal almost contact structures on Riemannian manifolds.

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associ…

We consider invariant symplectic connections $\nabla$ on homogeneous symplectic manifolds $(M,ω)$ with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible…

Noting that the complete lift of a Rimannian metric defined on a differentiable manifold is not 0-homogeneous on the fibers of the tangent bundle . In this paper we introduce a new lift which is 0-homogeneous. It determines on slit tangent bundle a pseudo-Riemannian metric, which depends only on the metric . We study s…

We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are $S^3 \times S^3$, the com…

We consider the evolution of an almost Hermitian metric by the $(1,1)$ part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric…

Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.

Classified spaces in low dimensions.

The paper examines the geometry of specific submanifolds in flag manifolds.

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

The paper constructs complex structures on specific manifolds using isoparametric theory.

We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromov's Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of t…

New manifolds found with almost everywhere positive curvature.

Survey updates knowledge on homogeneous Einstein spaces.

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic functions are define…

We discuss the complex geometry of two complex five-dimensional Kähler manifolds which are homogeneous under the exceptional Lie group $G_2$. For one of these manifolds rigidity of the complex structure among all Kählerian complex structures was proved by Brieskorn, for the other one we prove it here. We relate the Käh…

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly Kähler manifolds. Besides its almost complex structu…

We classify homogeneous pseudo-Riemannian manifolds of index 4 which admit an invariant almost hyper-Hermitian structure and an H-irreducible isotropy group. The main result is that all these spaces are flat except in dimension 12.

We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.

Study of complex surfaces in a specific pseudo-Riemannian space.

Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $φ$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $Aφ+φA=0$ if and only if it is …

Unique domain found in Einstein universe, simplifying manifold classification.

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…

Affine vector fields on pseudo-Kähler manifolds are symplectic.

Compact pseudo-Hermitian spaces have rigid holomorphic isometries.

An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…

The goal of this article is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group $H$ of isometries. In a first result, $H$ is a normal subgroup of the group of symmetries associated to the reducing tensor $\bar{S}$. The situation when $H$ is any group acting freely is an…

Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.

Consider a closed manifold $M$ immersed in $\R^m.$ Suppose that the trivial bundle $M\times\R^m=TM\otimes νM$ is equipped with an almost metric connection $\tilde{\nabla}$ which almost preserves the decomposition of $M\times\R^m$ into the tangent and the normal bundle. Assume moreover that the difference $Γ=\partial-\t…

Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost Kähler structure $(ω,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $ρ= λω$ for some $λ\in\mathbb R$, where $ρ$ is the Ricci…

Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kähler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, incl…

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …

The paper studies the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.

The article finds non-Abelian DT-instantons on non-Kähler manifolds.

In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order…

It is proved that if S^6 possesses an integrable complex structure, then there exists a 1-dimensional family of pairwise different exotic complex structures on P_3(C). This follows immediately from the main result of the paper: S^6 is not the underlying differentiable manifold of an almost homogeneous complex manifold …

The paper analyzes systoles of complex projective spaces under various metrics.

Study Nijenhuis operators on Banach homogeneous spaces, extending previous work.

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

Characterizes homogeneous spaces with geometric structures using connections.

An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in Γ({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of …